Method of Estimating a Leakage Current in a Semiconductor Device

ABSTRACT

In a method of estimating a leakage current in semiconductor device, a chip including a plurality of cells is divided into segments by a grid model. Spatial correlation is determined as spatial correlation between process parameters concerned with the leakage currents in each of the cells. A virtual cell leakage characteristic function of the cell is generated by arithmetically operating actual leakage characteristic functions. A segment leakage characteristic function is generated by arithmetically operating the virtual cell leakage characteristic functions of each cell in the segment. Then, a full chip leakage characteristic function is generated by statistically operating the segment leakage characteristic functions of each segment in the chip. Accordingly, the computational loads of Wilkinson&#39;s method for generating the full chip leakage characteristic function may be remarkably reduced.

REFERENCE TO PRIORITY APPLICATION

This application claims priority to Korean Patent Application No.2008-84718, filed Aug. 28, 2008, the contents of which are herebyincorporated herein in its entirety.

FIELD OF THE INVENTION

Example embodiments relate to a method of estimating a leakage currentin a semiconductor device, and more particularly, to a method ofsimulating and estimating a full chip leakage current for integratedcircuit designs in a semiconductor device.

BACKGROUND

As degrees of integration of semiconductor devices are increased and thesizes of semiconductor devices are decreased, serious problems withregard to the generation of leakage current are increasing. Thus,leakage estimation and reduction techniques have been one of the mostimportant design factors in manufacturing an integrated circuit. Theincreasing amounts of leakage current not only prevent normal operationof the integrated circuit but also consume excessive driving power, andthus it is increasingly critical for the device performance of theintegrated circuit. Particularly, battery-powered devices includingintegrated circuits, such as mobile and handheld electronics, haverecently become widespread, and thus the excessive power consumption hasbeen a critical factor for the performance of the battery-powereddevices. For those reasons, the leakage estimation and reductiontechniques have become much more important factors in designing the ICs.

Most of the leakage estimation and reduction techniques have focused onsub-threshold leakage due to lowering of power supply voltages andaccompanying reductions of threshold voltages. However, with the recenthigh degrees of integration and the decreases in the sizes of ICs, thegate leakage current occurring at a gate electrode of the ICs, as wellas the sub-threshold leakage current, has also become important factorsin designing the ICs. Accordingly, accurate full chip leakage estimationis frequently required for a chip design of the ICs so as to estimateboth of the sub-threshold leakage and the gate leakage. Particularly, inrecent very large-scale integration (VLSI) chips, tunneling of carriersthrough a gate insulation layer may frequently occur due to a reducedthickness of the gate insulation layer, and thus the estimation of thegate leakage necessarily needs to be considered in designing the VLSIchips.

Various estimation models for estimating the full chip leakage have beensuggested for the last few years. It is well known that the full chipleakage in a chip may be influenced by various factors, such as processparameters, for example, a line width and a critical dimension (CD) of apattern, and environmental factors, for example, channel temperature,power supply voltage (Vdd), circuit topology and an allowable load.Thus, a specific estimation model for estimating the full chip leakagewith respect to a specific factor does not give sufficiently accurateinformation on the full chip leakage in a chip.

Therefore, statistical models have been suggested for the full chipleakage estimation model in which conventional experimental results areoperated by well-known statistical methods and the full chip leakage isestimated in consideration of all of the above factors including theprocess parameters and the environmental factors. Particularly, a lognormal estimation model has been most widely used for the full chipleakage estimation model among the above statistical models. Accordingto the lognormal estimation model, every factor of the processparameters and the environmental factors functions as a random variablefor a probability density function (PDF), and thus a lognormal PDF isgenerated with respect to each of the various process parameters and theenvironmental factors. Then, each of the lognormal PDFs is functionallysummarized into a single composite probability function, and an optimalpoint is determined by using the single composite probability functionwhere the full chip leakage is minimized in consideration of all of theabove factors. The values of the random variables at the optimal pointof the single composite probability function are regarded as optimaldesign factors for minimizing the full chip leakage in a chip. Therandom variable of the lognormal PDF is an exponential of the randomvariable of a normal PDF, and thus the multiplication of the lognormalrandom variables is also distributed in accordance with a lognormaldistribution. For those reasons, the lognormal variable has been widelyused for statistical estimation and analysis for a situation where thestatistical error is strongly influenced by multiplication ofenvironmental factors.

However, the statistical estimation model for the full chip leakageusing the lognormal PDF (hereinafter referred to as lognormal leakageestimation model) has a critical demerit of computational complexity.Particularly, the computational complexity of the lognormal leakageestimation model may be geometrically increased as the number of theenvironmental factors related to the statistical error of the lognormalleakage estimation model. Thus, the lognormal leakage estimation modelhas many limitations when being applied to a circuit design.

According to a conventional lognormal leakage estimation model, asemiconductor chip is divided into a plurality of estimation regions bya grid and the lognormal PDF is generated at each of the estimationregions. Then, each of the lognormal PDFs at the estimation regions isstatistically summed up in consideration of spatial correlation betweenthe estimation regions, to thereby generate a full chip lognormal PDFindicating probability of a leakage current from a whole chip on thewafer.

Particularly, the probability of the leakage current at an arbitrarycell l of the chip is expressed as follows when the leakage current isinfluenced by an arbitrary environmental factor i.

$\begin{matrix}{I_{i}^{l} = ^{a_{0} + {\sum\limits_{j = 1}^{n}{a_{j}P_{j}}} + {a_{n + 1}R}}} & (1)\end{matrix}$

The polynomial term in the above exponential equation (1) indicates anormal distribution having an average of a₀ and a variation of Σa_(j) ²on condition that the parameters P_(j) and R may be distributed as astandard normal distribution N(0,1) Accordingly, the leakage current atan arbitrary cell is calculated by equation (1) on a whole chip, andthus is distributed on the whole chip as a lognormal distribution.

In the above equation (1), the parameter P_(j) is a global parameterindicating outer environment factors at each of the chips. That is, theparameter P_(j) includes a random variable indicating random variationin a manufacturing process for a semiconductor device. The randomvariable of the individual chip includes a variation of a die-to-dieparameter in which the parameter may be uniformly varied on the wholechip and a variation of a within-die-parameter in which the parametermay be non-uniformly varied on the whole chip, and thus the variation ofthe within-die-parameter may be expressed by spatial correlation.

In addition, the parameter R in the above equation (1) is a localparameter as a single random variable into which various independentvariables are grouped. The independent variable has an individual effecton the current leakage at a local area of the chip irrespectively ofother variables. A plurality of the independent variables can be treatedas a single variable without any computational error in equation (1),and thus the number of the variables is remarkably reduced in conductingequation (1) to thereby significantly reduce the computational load.

Further, a₀, a_(j), a_(n+1), in equation (1) are fitting coefficientsfor transforming the distribution of the probability function expressedby equation (1) into a normal distribution and indicate correlationbetween the global parameter or the local parameter and the leakagecurrent. When the probability function of equation (1) including theglobal parameter and the local parameter is transformed into the normaldistribution, multiplication of the lognormal distribution istransformed into summation of the standard normal distribution, tothereby significantly reduce the computational load when performing aleakage estimation process.

Since the leakage current at an arbitrary cell is calculated by equation(1) with respect to a specific input factor, an average leakage currentover a whole chip (full chip leakage current) with respect to all of theinput factors may be calculated by the following equation (2).

$\begin{matrix}{I_{avg} = {{\sum\limits_{l = 1}^{p}\left( {\sum\limits_{i = 1}^{m}{\Pr_{i}I_{i}^{l}}} \right)} = {\sum\limits_{l = 1}^{p}\left( {\sum\limits_{i = 1}^{m}{\Pr_{i}^{a_{0} + {\sum\limits_{j = 1}^{n}{a_{j}p_{j}}} + {a_{n + 1}R}}}} \right)}}} & (2)\end{matrix}$

In the above equation (2), Pr_(i) indicates a probability that anarbitrary input factor i may be applied to an arbitrary cell l of thechip and the number of the input factors is mi and number of the cellson the chip is p.

As shown in the above equation (2), the full chip leakage current iscalculated through the summation of the lognormal distribution andWilkinson's method has been widely used for the summation of thelognormal distribution.

According to Wilkinson's method, a single lognormal distribution, whichmay be statistically equivalent to the summation of a number of theindividual lognormal distributions, is generated by using a first momentand a second moment as follows.

e ^(Y) ¹ +e ^(Y) ² + . . . +e ^(Y) ^(n) =e ^(Z)   (3)

In the above equation (3), each of the probability distributions Y_(i)includes a normal distribution having an average of μ_(Y) _(i) and astandard variation of σ_(Y) _(i) . Supposing that the probabilitydistribution Z of the equivalent lognormal probability function is anormal distribution having an average of μ_(z) and a standard variationof σ_(z), the first and second moments of the equivalent lognormaldistribution are expressed as follows.

$\begin{matrix}{\mu_{1} = {{E\left\lbrack {^{Y_{1}} + ^{Y_{2}} + \ldots + ^{Y_{n}}} \right\rbrack} = ^{\mu_{z} + \frac{\sigma_{2}^{2}}{2}}}} & (4) \\\begin{matrix}{\mu_{2} = {E\left\lbrack \left( {^{Y_{1}} + ^{Y_{2}} + {\ldots \mspace{14mu} ^{Y_{n}}}} \right)^{2} \right\rbrack}} \\{= ^{{2\mu_{z}} + {2\sigma_{z}^{2}}}} \\{= {{\sum\limits_{i = 1}^{n}^{{2\mu_{Yi}} + {2\sigma_{Yi}^{2}}}} + {2{\sum\limits_{i = 1}^{n - 1}\left( {\sum\limits_{j = {i + 1}}^{n}^{\mu_{Yi} + \mu_{Yj} + \frac{\begin{matrix}{\sigma_{Yi}^{2} + \sigma_{Yj}^{2} +} \\{2r_{ij}\sigma_{Yi}\sigma_{Yj}}\end{matrix}}{2}}} \right)}}}}\end{matrix} & (5)\end{matrix}$

In the above equation (5), r_(ij) indicates a correlation coefficientbetween the different probability distributions Y_(i) and Y_(j).

Solutions of the simultaneous equations (4) and (5) provide the averageand the standard deviation of the probability distribution Z, to therebydetermine the lognormal distribution Z. Then, the full chip leakagecurrent is estimated by the lognormal distribution Z.

However, as shown in the second term of equation (5), the correlationcoefficients between the different probability distributions usuallycause a tremendous computational load in operating the statisticalsummation of equation (5), to thereby significantly increase theoperation complexity of equation (5).

The operation complexity of the lognormal distribution model isdetermined by the number of the individual lognormal distributions andis expressed as O(N²). N is the number of the individual lognormaldistributions that are to be statistically summed up in calculating thefirst and second moments. According to the conventional lognormaldistribution model for estimating the full chip current leakage, anumber of (NC*M) of the independent lognormal distributions aregenerated when the full chip includes a number of NC of the cells andthe kinds of the current leakage is a number of M and the operationcomplexity of Wilkinson's method is calculated as O((NC*M)²). Therefore,the operation complexity is extremely increased when Wilkinson's methodis performed on a relatively large size of an electric circuit andfinally exceeds the operation capability of the current computersystems. Particularly, when the degree of integration of semiconductordevices is increased, and thus a plurality of electric circuits isintegrated into a small area of a substrate, the operation complexity ofWilkinson's method is further increased and finally Wilkinson's methodcannot be substantially applied for estimating the full chip leakage.

For the above reasons, there is still a need for an improved estimationmodel for estimating the full chip leakage current in which the fullchip leakage current is accurately estimated with a relatively lowoperation complexity.

SUMMARY

Example embodiments provide a method of estimating a full chip leakagecurrent for a semiconductor device.

According to some example embodiments, there is provided a method ofestimating a leakage current in semiconductor device. At first, a chipon a substrate may be divided into a number of segments. The chip mayinclude a plurality of cells on which various unit conductive structuresof an integrated circuit is formed. Spatial correlation may bedetermined between process parameters that are concerned with theleakage currents in each of the cells. A virtual cell leakagecharacteristic function of the cell may be generated by arithmeticallyoperating actual leakage characteristic functions that determine theleakage currents generated from the cell, respectively. The virtual cellleakage characteristic function may be equivalent to actual leakagecharacteristic functions and a virtual leakage current generated by thevirtual cell leakage characteristic function may be equivalent with theleakage currents in the cell. A segment leakage characteristic functionmay be generated by arithmetically operating the virtual cell leakagecharacteristic functions of each cell in the segment. The segmentleakage characteristic function may determine a virtual leakage currentgenerated from all of the segments of the chip. A full chip leakagecharacteristic function may be generated by statistically operating thesegment leakage characteristic functions of each segment in the chip.The full chip leakage characteristic function may determine a virtualleakage current generated from the whole chip of the semiconductordevice.

In an example embodiment, the actual leakage characteristic function andthe virtual cell leakage characteristic function is expressed as anexponential polynomial with respect to the process parameter.

For example, the actual leakage characteristic function may include afirst probability density function (PDF) determining a first leakagecurrent caused by the process parameter and expressed as equation (1)and a second PDF determining a second leakage current caused by theprocess parameter and expressed as equation (2),

e^(ƒ) ¹ ^((P) ¹ ^(,P) ² ^(. . . ,P) ^(n) ⁾   (1),

e^(ƒ) ² ^((P) ¹ ^(,P) ² ^(. . . ,P) ^(n) ⁾   (2).

The virtual cell leakage characteristic function may include a third PDFthat equals arithmetic summation of the first and the second PDFs, andis thus expressed as equation (3),

e^(ƒ) ³ ^((P) ¹ ^(,P) ² ^(. . . ,P) ^(n) ⁾=e^(ƒ) ¹ ^((P) ¹ ^(,P) ²^(. . . ,P) ^(n) ⁾+e^(ƒ) ² ^((P) ¹ ^(,P) ² ^(. . . ,P) ^(n) ⁾  (3)

(wherein the small capital letters e in the above equations indicates anatural log and pi indicates the process parameter that causes theleakage current in the cell).

In an example embodiment, an exponential term of the first PDF mayinclude a polynomial of a normal distribution having an average of a_(o)and a variation of

$\sum\limits_{j = 1}^{n + 1}a_{j}^{2}$

so that the first PDF may be expressed as equation (4) indicating alognormal distribution,

$\begin{matrix}{^{a_{0} + {\sum\limits_{j = 1}^{n}{a_{j}P_{j}}} + {a_{n - 1}R}}.} & (4)\end{matrix}$

An exponential term of the second PDF includes a polynomial of a normaldistribution having an average of b₀ and a variation of

$\sum\limits_{j = 1}^{n + 1}b_{j}^{2}$

so that the second PDF is expressed as equation (5) indicating alognormal distribution,

$\begin{matrix}{^{b_{0} + {\sum\limits_{j = 1}^{n}{b_{j}P_{j}}} + {b_{n - 1}R}}.} & (5)\end{matrix}$

An exponential term of the third PDF may include a polynomial of anormal distribution having an average of c₀ and a variation of

$\sum\limits_{j = 1}^{n + 1}c_{j}^{2}$

so that the second PDF is expressed as equation (6) indicating alognormal distribution,

$\begin{matrix}{^{c_{0} + {\sum\limits_{j = 1}^{n}{c_{j}P_{j}}} + {c_{n - 1}R}},} & (6)\end{matrix}$

on condition that

${c_{0} = {{2\; {\ln \left( M_{1} \right)}} - {\frac{1}{2}{\ln \left( M_{2} \right)}}}},{{\sum\limits_{j = 1}^{n + 1}c_{j}^{2}} = {{\ln \left( M_{2} \right)} - {2\; {\ln \left( M_{1} \right)}}}},{c_{j} = \frac{{{E\left( ^{f_{1}} \right)}^{\sum\limits_{j = 1}^{n}{a_{j}k_{j}}}a_{j}} + {{E\left( ^{f_{2}} \right)}^{\sum\limits_{j = 1}^{n}{b_{j}k_{j}}}b_{j}}}{{E\left( ^{f_{3}} \right)}^{\sum\limits_{j = 1}^{n}{c_{j}k_{j}}}}},{and}$${c_{n + 1} = \sqrt{\sigma_{c}^{2} - {\sum\limits_{j = 1}^{n}c_{j}^{2}}}},$

wherein a, b and c denotes a fitting coefficient of a normaldistribution, M1 and M2 denotes first and second moments of the thirdPDF, R denotes a local parameter by which the first and the secondleakage currents are generated independently from each other and Pdenotes a global parameter by which both of the first and the secondleakage currents are commonly generated.

The first and second moments of the third PDF may be statisticallyobtained as follows:

M ₁ =E[e ^(ƒ) ¹ +e ^(ƒ) ² ], M ₂ =E[(e ^(ƒ) ¹ +e ^(ƒ) ² )²].

The fitting coefficient c_(j) is obtained through the following steps:firstly, a second moment equivalence condition of a lognormaldistribution composition may be applied on condition that an arbitrarylognormal distribution e^(Z) is added to right and left-hand sides ofequation (3), to thereby obtain an equation (7) as follows:

$\begin{matrix}{{{E\left\lbrack \left( {^{f_{3}} + ^{Z}} \right)^{2} \right\rbrack} = {{{E\left\lbrack \left( {\left( {^{f_{1}} + ^{f_{2}}} \right) + ^{Z}} \right)^{2} \right\rbrack}->{E\left\lbrack {^{f_{3}}^{Z}} \right\rbrack}} = {{{E\left\lbrack {{^{f_{1}}^{Z}} + {^{f_{2}}^{Z}}} \right\rbrack}->{{E\left\lbrack ^{f_{3}} \right\rbrack}^{\sum\limits_{j = 1}^{n}{c_{j}z_{j}}}}} = {{{E\left\lbrack ^{f_{1}} \right\rbrack}^{\sum\limits_{j = 1}^{n}{a_{j}z_{j}}}} + {{E\left\lbrack ^{f_{2}} \right\rbrack}^{\sum\limits_{j = 1}^{n}{b_{j}z_{j}}}}}}}};} & (7)\end{matrix}$

Then, equation (7) may be expanded into a first order Taylor series andthe Taylor series of equation (7) may expanded into an identicalequation with respect to an arbitrary random variable z of the arbitrarylognormal distribution e^(Z).

In an example embodiment, the global parameter may includes a chip-basedvariable which may be involved with the leakage current by the chip andan inner-chip variable having spatial correlation between the leakagecurrents in the chip and the local parameter may include the variableshaving no spatial correlation between the leakage currents in the chip.

In an example embodiment, the first and the second leakage currents mayinclude one of a sub-threshold leakage current and a gate leakagecurrent, respectively.

In an example embodiment, the virtual cell leakage characteristicfunction may includes a PDF of a lognormal distribution of which theexponential term is a polynomial of a normal distribution having anaverage of c₀ and a variation of

$\sum\limits_{j = 1}^{n + 1}c_{j}^{2}$

so that the virtual cell leakage characteristic function is expressed asequation (8),

$\begin{matrix}^{c_{0} + {\sum\limits_{j = 1}^{n}{c_{j}P_{j}}} + {c_{m + 1}R}} & (8)\end{matrix}$

(wherein, R denotes a local parameter having no spatial correlationbetween the cells in the segment and m denotes a number of the processparameters that are not treated as the local parameter in the segment).Then, the segment leakage characteristic function may be generated byarithmetically adding the virtual cell leakage characteristic functionexpressed by equation (8) at every cell in the segment. The segmentleakage characteristic function may include a PDF of a lognormaldistribution of which the exponential term may be a polynomial of anormal distribution.

In an example embodiment, the virtual cell leakage characteristicfunction may be arithmetically added to each other as follows. A secondmoment equivalence condition of a lognormal distribution composition maybe applied in Wilkinson's method, to thereby obtain an exponentialpolynomial equation. Then, the exponential polynomial equation may beexpanded into a first order Taylor series, and the Taylor series of theexponential polynomial equation may be into an identical equation withrespect to a random variable of an arbitrary lognormal distribution. Insuch a case, the segment leakage characteristic function may include oneof a sub-threshold leakage current and a gate leakage current.

In an example embodiment, the step of the generating the full chipleakage characteristic function may include obtaining an average and avariation by using first and second moments of a number of the segmentleakage characteristic functions.

In an example embodiment, the actual leakage characteristic function maybe obtained by analyzing experimental data including the leakagecurrents and the process parameters. For example, the actual leakagecharacteristic function may be obtained by a regression analysis processonto the experimental data, so that a statistical relation between theleakage current and the process parameter is generated.

In an example embodiment, the process parameter may include a globalparameter having a chip-based variable which may be involved with theleakage current by the chip and an inner-chip variable which has spatialcorrelation between the leakage currents in the chip and the localparameter includes the variables having no spatial correlation betweenthe leakage currents in the chip.

In an example embodiment, the process parameter may include a randomparameter having relation to a random variation that may be randomlycaused by environmental factors in performing a process and a systematicparameter having relation to a systematic variation that may be causedby physical factors of a process system for performing the process.

In an example embodiment, the random variation may be expressed as a PDFhaving the random parameter as a variable which may determine aprobability distribution and the systematic variation may be expressedas a spatial correlation matrix.

In an example embodiment, the process parameter may include one of atemperature of a deposition process, a thickness of a deposited layer, apattern width and a gate voltage.

In an example embodiment, a variation analysis may be further performedby arithmetically operating the virtual cell leakage characteristicfunction and a supplemental leakage characteristic function that maydetermine a supplemental leakage current generated from the cell, tothereby analyze variation of the virtual cell leakage characteristicfunction due to the supplemental leakage characteristic function.

According to some example embodiments, lognormal distributions about aleakage characteristic function may be summed up not by a statisticalprocess but by an arithmetic process using an exponential polynomial ofthe PDF of a lognormal distribution. Therefore, the complexity ofWilkinson's method for generating a full chip leakage characteristicfunction may be remarkably reduced without any deterioration of theaccuracy of the full chip leakage characteristic function.

In addition, when performing a principal component analysis (PCA) on awafer chip, a computational overload for processing a spatialcorrelation matrix may be sufficiently minimized. Furthermore, avariation analysis for analyzing the effect of a variation of an actualleakage characteristic function on a virtual cell leakage characteristicfunction, which may be known as an incremental analysis in theconventional Wilkinson's method, may be performed merely by a series ofarithmetic operations, to thereby sufficiently reduce the computationalloads of the conventional Wilkinson's method.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart showing a method of estimating a full chip leakagecurrent for a semiconductor device;

FIG. 2 is a plan view illustrating a chip on a wafer on which aplurality of segments is formed in accordance with a grid model; and

FIG. 3 is a flowchart showing processing steps for generating thevirtual cell leakage characteristic function shown in FIG. 1.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Various example embodiments will be described more fully hereinafterwith reference to the accompanying drawings, in which some exampleembodiments are shown. The present invention may, however, be embodiedin many different forms and should not be construed as limited to theexample embodiments set forth herein. Rather, these example embodimentsare provided so that this disclosure will be thorough and complete, andwill fully convey the scope of the present invention to those skilled inthe art. In the drawings, the sizes and relative sizes of layers andregions may be exaggerated for clarity.

It will be understood that when an element or layer is referred to asbeing “on,” “connected to” or “coupled to” another element or layer, itcan be directly on, connected or coupled to the other element or layeror intervening elements or layers may be present. In contrast, when anelement is referred to as being “directly on,” “directly connected to”or “directly coupled to” another element or layer, there are nointervening elements or layers present. Like numerals refer to likeelements throughout. As used herein, the term “and/or” includes any andall combinations of one or more of the associated listed items.

It will be understood that, although the terms first, second, third,etc. may be used herein to describe various elements, components,regions, layers and/or sections, these elements, components, regions,layers and/or sections should not be limited by these terms. These termsare only used to distinguish one element, component, region, layer orsection from another region, layer or section. Thus, a first element,component, region, layer or section discussed below could be termed asecond element, component, region, layer or section without departingfrom the teachings of the present invention.

Spatially relative terms, such as “beneath,” “below,” “lower,” “above,”“upper” and the like, may be used herein for ease of description todescribe one element or feature's relationship to another element(s) orfeature(s) as illustrated in the figures. It will be understood that thespatially relative terms are intended to encompass differentorientations of the device in use or operation in addition to theorientation depicted in the figures. For example, if the device in thefigures is turned over, elements described as “below” or “beneath” otherelements or features would then be oriented “above” the other elementsor features. Thus, the exemplary term “below” can encompass both anorientation of above and below. The device may be otherwise oriented(rotated 90 degrees or at other orientations) and the spatially relativedescriptors used herein interpreted accordingly.

The terminology used herein is for the purpose of describing particularexample embodiments only and is not intended to be limiting of thepresent invention. As used herein, the singular forms “a,” “an” and“the” are intended to include the plural forms as well, unless thecontext clearly indicates otherwise. It will be further understood thatthe terms “comprises” and/or “comprising,” when used in thisspecification, specify the presence of stated features, integers, steps,operations, elements, and/or components, but do not preclude thepresence or addition of one or more other features, integers, steps,operations, elements, components, and/or groups thereof.

Example embodiments are described herein with reference tocross-sectional illustrations that are schematic illustrations ofidealized example embodiments (and intermediate structures). As such,variations from the shapes of the illustrations as a result, forexample, of manufacturing techniques and/or tolerances, are to beexpected. Thus, example embodiments should not be construed as limitedto the particular shapes of regions illustrated herein but are toinclude deviations in shapes that result, for example, frommanufacturing. For example, an implanted region illustrated as arectangle will, typically, have rounded or curved features and/or agradient of implant concentration at its edges rather than a binarychange from implanted to non-implanted region. Likewise, a buried regionformed by implantation may result in some implantation in the regionbetween the buried region and the surface through which the implantationtakes place. Thus, the regions illustrated in the figures are schematicin nature and their shapes are not intended to illustrate the actualshape of a region of a device and are not intended to limit the scope ofthe present invention.

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which this invention belongs. It will befurther understood that terms, such as those defined in commonly useddictionaries, should be interpreted as having a meaning that isconsistent with their meaning in the context of the relevant art andwill not be interpreted in an idealized or overly formal sense unlessexpressly so defined herein.

Hereinafter, example embodiments will be explained in detail withreference to the accompanying drawings.

FIG. 1 is a flowchart showing a method of estimating a full chip leakagecurrent for a semiconductor device. FIG. 2 is a plan view illustrating achip on a wafer on which a plurality of segments is formed in accordancewith a grid model.

Referring to FIGS. 1 and 2, a wafer chip 100 may be divided into aplurality of segments for estimating a full chip leakage current inaccordance with an example embodiment of the present inventive concept(step S100). The wafer chip may include a plurality of cells on whichvarious conductive structures for integrated circuits are arranged.

In an example embodiment, the wafer chip 100 may include a DRAM devicehaving a transistor and a capacitor as an operating unit and a flashmemory device having a selection transistor, a cell transistor and aground transistor, which are arranged in a line, as an operating unit. Asurface of the wafer chip 100 may be divided into a plurality ofsegments A by a virtual grid and a plurality of cells may be positionedin each of the segments A. A plurality of the operating units may bepositioned in each of the cells.

In the present example embodiment, the wafer chip 100 may be dividedinto nine segments A and each of the segments A may be designated by amatrix index. Various conductive structures may be formed on each of thesegments A in accordance with manufacturing process on a wafer. Forexample, first and second conductive structures C1 and C2, which aredifferent from each other, may be formed on a first segment A₁₁ and athird and a fourth conductive structures C3 and C4 may be formed on afifth segment A₂₂ and a ninth segment A₃₃, respectively. The thirdand/or fourth conductive structures C3 and C4 may be substantiallyidentical to the first and/or second conductive structures C1 and C2, aswould be known to one of ordinary skill in the art.

A local leakage current may be generated from each of the segments A ofthe wafer chip 100 and a total current leakage generated from the waferchip 100 (full chip leakage current) may be estimated by the followingprocess.

At first, various process variations, which may cause the local leakagecurrent at each cell, may be detected and every process parametercorresponding to the process variation may be found at each of thecells. Then, a spatial correlation may be determined between the processparameters of the same process variation on the wafer chip 100 (stepS200).

In an example embodiment, various pattern variations may be generated byvarious factors in a manufacturing process for a semiconductor deviceand the leakage current in a transistor may be influenced by the patternvariations. For example, a size or a shape of a pattern may be changedby lens distortion in an exposure process or variations of processconditions in an etching process. The pattern variations over anallowable range may cause process failures in each unit process of themanufacturing process. The pattern variations in the manufacturingprocess for a semiconductor device may include a random variation causedby variations of process conditions and a systematic variation caused bycharacteristics and specifications of systems or equipment forperforming the unit process.

The random variation indicates a kind of the pattern variation that maybe arbitrarily generated in accordance with process conditions and waferconditions, and thus a statistical estimation model having each ofprocess parameters of a specific unit process as a random variable mayprovide an accurate performance variation of a semiconductor device inrelation to the process parameter. For example, the leakage current dueto the random variation may be described by a probability densityfunction (PDF) with respect to the random variable in relation to therandom variation.

The systematic variation indicates a kind of pattern variation that maybe generated due to systematic factors of physical equipment forperforming the process and is a relative variation of the pattern at alocal area with respect to a reference local area. For example, thesystematic factors of physical equipment may include instrumentaleffects on a respective layout and position variation of a wafer in theequipment. Therefore, the systematic variation may be numericallyquantified by a spacious correlation based on each position function ofthe local area of the wafer, not by a unique PDF with respect to aspecific process parameter in case of the random variation.

For example, although the second conductive structure C2 on the firstsegment A11 may be the same as the third conductive structure on thefifth area A22, the probability of the pattern variations may bedifferent from each other because the first and fifth conductivestructures may be positioned at different area of the wafer chip 100.Thus, the probability difference of the pattern variation of eachconductive structure at each of the segments and statistical correlationbetween the same conductive structures at different segments may bedetermined on the wafer chip 100, to thereby obtain the spaciouscorrelation between the process parameters.

For example, the process parameter may include a deposition temperatureand a thickness of a deposited layer in a deposition process, a linewidth of a pattern and a voltage of a gate electrode (Vdd).

Then, an actual leakage characteristic function indicating a PDF of aleakage current generated from a particular structure may be determinedin a cell of the wafer chip 100, to thereby generate various actualleakage characteristic functions at the cell. Thereafter, the variousactual leakage characteristic functions may be statistically summed upto generate a virtual cell leakage characteristic function indicating aPDF of a total leakage current generated from the cell of the wafer chip100 (step S300).

Various actual leakage currents may be generated at each of the cells ofthe wafer chip 100 in accordance with mechanical characteristics ofleakage and processing limitations and the processing parameterscorresponding each of the actual leakage currents may be the same ordifferent from each other.

A particular process parameter, which may be involved with a specificactual leakage current and have no effect on the other actual leakagecurrents, may be defined as a local variable and another particularprocess parameter, which may be commonly involved with all of the actualleakage currents, may be defined as a global variable. The globalvariables may include a chip-based variable which may be involved withthe actual leakage current by the chip and an inner-chip variable havingspatial correlation between the actual leakage currents in the chip. Thelocal variables may include all of the inner-chip variables and theouter-chip variables having no spatial correlation between the actualleakage currents in the chip.

FIG. 3 is a flowchart showing processing steps for generating thevirtual cell leakage characteristic function shown in FIG. 1.

In an example embodiment, an object wafer on which various conductivestructures and patterns may have been formed in a manufacturing processmay be inspected and various experimental data indicating correlationbetween each of the process parameters and each of the actual leakagecurrent may be obtained (step S310) by various well-known inspectionprocesses. Then, the experimental data may be statistically processed tothereby generate an exponential polynomial as the actual leakagecharacteristic function (step S320).

For example, repeated inspection processes may be performed on anexperimental chip for inspecting the leakage current characteristics andvarious experimental data indicating the correlation between the actualleakage current and the process parameter may be obtained and stored ina predetermined data structure. Then, the data structure may bestatistically treated by a statistical model to thereby generate theactual leakage characteristic function in a circuit design stage.

The data structure of the correlation between the actual leakage currentand the process parameter may be statistically treated by a regressionanalysis method, and thus the actual leakage characteristic function maybe expressed as a PDF with respect to the actual current leakage(hereinafter referred to as actual leakage PDF).

Accordingly, the actual current leakage may be generated at a particularcell of a specific segment of the wafer chip 100 in accordance with alognormal distribution defined by a PDF with respect to the above localand global variables as the following equation (1).

$\begin{matrix}^{a_{0} + {\sum\limits_{j = 1}^{n}{a_{j}P_{j}}} + {a_{n + 1}R}} & (1)\end{matrix}$

The polynomial expression in the PDF of equation (1) may indicatestatistical correlation between the process parameter and a specificactual leakage current at a specific cell through the regressionanalysis method. The random variables of P and R in the above equation(1) may be varied in accordance with a normal distribution having anaverage of a₀ and a standard variation of

${\sum\limits_{j = 1}^{n + 1}a_{j}^{2}},$

and the coefficient a in the above equation (1) may include aneigenvalue for fitting the actual leakage current to a respectiveprocess parameter (hereinafter referred to as fitting coefficient).

That is, the actual leakage characteristic function may be a PDF of alognormal distribution since the exponential term of equation (1) maydefine the normal distribution.

Various actual leakage currents may be measured from the segment of thewafer chip 100 and the actual leakage characteristic functions may begenerated with respect to each of the actual leakage currents,respectively in every feasible cell of the same segment through theregression analysis method. Thus, a number of the actual leakagecharacteristic functions may be generated with respect to every actualleakage current in the feasible cells of the wafer chip 100.

In the present example embodiment, the virtual leakage characteristicfunction may be generated by the following process supposing that twokinds of the actual leakage currents are measured in the same segment ofthe wafer chip 100. However, when three or more kinds of the actualleakage currents may be measured in the same segment, the virtualleakage characteristic function may also be generated by the sameprocess, as would be known to one of the ordinary skill in the art.

A first leakage current may be distributed in accordance with a firstPDF as described in the above equation (1) and a second leakage currentmay be distributed in accordance with a second PDF as described in thefollowing equation (2).

$\begin{matrix}^{b_{0} + {\sum\limits_{j = 1}^{n}{b_{j}P_{j}}} + {b_{n + 1}R}} & (2)\end{matrix}$

The exponential term in the second PDF of equation (2) may define anormal distribution having an average of b₀ and a standard variation of

${\sum\limits_{j = 1}^{n + 1}b_{j}^{2}},$

and the coefficient b may be the fitting coefficient of the second PDF.Accordingly, the actual leakage characteristic function corresponding tothe second actual leakage current may be a PDF of a lognormaldistribution since the exponential term of equation (2) may define thenormal distribution.

Then, each of the actual leakage characteristic functions correspondingto the first and second actual leakage current may be operatedarithmetically, not statistically, to thereby generate a third PDF as avirtual cell leakage characteristic function with respect to thespecific cell of the segment (step S330).

Supposing that the first, second and third PDF is described as e^(A),e^(B) and e^(C), respectively, the virtual cell leakage characteristicfunction may be approximated to the following equation (3).

e ^(A) +e ^(B) =e ^(C)   (3)

Accordingly, equation (3) may be described into the polynomial type asfollows by equation (3-1).

$\begin{matrix}{{^{a_{0} + {\sum\limits_{j = 1}^{n}{a_{j}P_{j}}} + {a_{n + 1}R}} + ^{b_{0} + {\sum\limits_{j = 1}^{n}{b_{j}P_{j}}} + {b_{n + 1}R}}} = ^{c_{0} + {\sum\limits_{j = 1}^{n}{c_{j}P_{j}}} + {c_{n + 1}R}}} & \left( {3\text{-}1} \right)\end{matrix}$

The polynomials at the exponential terms of equation (3) may be treatedarithmetically, not statistically, as will be described hereinafter tothereby reduce the computational load as compared with the conventionalWilkinson's method. Determination of the coefficient c_(j) in the aboveequation (3-1) requires (n+2) simultaneous equations. Among the (n+2)equations, two equations may be obtained by a statistical method and therest of the n equations may be obtained by a correlation analysisbetween the actual leakage currents. Then, a solution of the (n+2)simultaneous linear equations may determine the coefficient c_(j) in theabove equation (3-1), to thereby generate the third PDE

The above two equations obtained by a statistical method may be a firstmoment and a second moment of the third PDF described as follows.

An average and a variation of the third PDF may be μ_(c) and σ_(c),respectively, according to the following equations (4) and (5).

$\begin{matrix}{\mu_{c} = {{{2\; {\ln \left( M_{1} \right)}} - {\frac{1}{2}{\ln \left( M_{2} \right)}}} = c_{0}}} & (4) \\{\sigma_{c}^{2} = {{{\ln \left( M_{2} \right)} - {2\; {\ln \left( M_{1} \right)}}} = {\sum\limits_{j = 1}^{n + 1}c_{j}^{2}}}} & (5)\end{matrix}$

In the above equations (4) and (5), M₁ and M₂ indicate the first andsecond moments of the third PDF statistically obtained by Wilkinson'smethod, respectively.

So as to obtain the other n equations, an arbitrary supplementallognormal distribution with respect to an arbitrary actual leakagecurrent may be added at the right-hand side and the left-hand side ofequation (3). When two arbitrary lognormal distributions is summed up,the summation process of the present example embodiment may besubstantially the same as the conventional Wilkinson's method. However,when another lognormal distribution may be added to the summationresults of the two lognormal distributions, the summation process of thepresent example embodiment may be clearly different from theconventional Wilkinson's method. The differentiation of the summationbetween the conventional Wilkinson's method and the present inventiveconcept lies in correlation between the right-hand-side supplementallognormal distribution and the left-hand-side supplemental lognormaldistribution in the above equation (3).

According to the conventional Wilkinson's method, two arbitrarylognormal distributions are statistically summed up to generate a firstcomposed lognormal distribution that is completely different from theoriginal lognormal distributions and is statistically equivalent to thesummation of the original lognormal distributions. Thus, the originallognormal distributions with respect to a process parameter iscompletely replaced by a new lognormal distribution, the first composedlognormal distribution, with respect to the same process parameter thatis statistically equivalent to the summation of the original lognormaldistributions. Adding the supplemental lognormal distribution to thefirst composed lognormal distribution may generate a second composedlognormal distribution that is completely different from the firstcomposed lognormal distribution and is statistically equivalent to thesummation of the supplemental lognormal distribution and the firstcomposed lognormal distribution. Therefore, an actual supplementalleakage current represented by the supplemental lognormal distributionmay have statistical correlation with a virtual equivalent leakagecurrent represented by the first composed lognormal distribution and maynot have statistical correlation with the actual leakage currentsrepresented by the original lognormal distributions.

For the above reasons, the second composed lognormal distribution at theleft-hand side of equation (3), i.e., the summation results of thesupplemental lognormal distribution to the left-hand side of equation(3), may not be statistically correlated with the summation results ofthe supplemental lognormal distribution to the right-hand side ofequation (3). That is, the second composed lognormal distribution at theleft-hand side of equation (3) may have operational errors in view ofthe summation results at the right-hand side of equation (3). So as toeliminate the operational errors in the second composed lognormaldistribution, the summation process for generating the second composedlognormal distribution needs to include the summation process forgenerating the first composed lognormal distribution. That is, thesummation of the lognormal distributions at the left-hand side ofequation (3) may be processed simultaneously, not sequentially or stepby step. As a result, the computational loads of the conventionalWilkinson's method may be tremendously increased according to anincrease of the number of the added lognormal distributions.

However, when equation (3) may come into existence irrespectively of thesupplemental lognormal distribution added to the right and left-handsides thereof, the equivalent lognormal distribution may be accuratelygenerated merely by arithmetic summation of the lognormal distributions,not by statistical summation of the lognormal distribution. Thearithmetic equivalent lognormal distribution may preserve thestatistical correlation between the actual leakage currents with respectto the process parameter, and thus sufficiently eliminate the operationerrors of the conventional Wilkinson's method.

Supposing that the supplementary lognormal distribution is representedas e^(z), the adding of the supplementary lognormal distribution toequation (3) may be expressed as the following equation (6).

(e ^(A) +e ^(B))+e ^(Z) =e ^(C) +e ^(Z)   (6)

So as to be statistically equal between the left and right-hand sides ofequation (6), the first and second moments at the left and right-handsides of equation (6) need equal to each other, respectively. The firstand second moments of equation (6) may be represented as the followingequations (7) and (8).

E[e ^(C) +e ^(Z) ]=E[e ^(Z) +e ^(B) ]+E[e ^(Z) ]=E[e ^(A) +e ^(B) +e^(C)]  (7)

E[(e ^(C) +e ^(Z))² ]=E[((e ^(A) +e ^(B))+e ^(Z))² ]→E[e ^(C) e ^(z)]=E[e ^(A) e ^(Z) +e ^(B) e ^(Z)]  (8)

Equation (8) is expressed as the following equation (9).

$\begin{matrix}{^{c_{0} + {{({\sum\limits_{j = 1}^{n + 1}c_{j}^{2}})}/2} + {\sum\limits_{j = 1}^{n}{c_{j}z_{j}}}} = {^{a_{0} + {{({\sum\limits_{j = 1}^{n + 1}a_{j}^{2}})}/2} + {\sum\limits_{j = 1}^{n}{a_{j}z_{j}}}} + ^{b_{0} + {{({\sum\limits_{j = 1}^{n + 1}b_{j}^{2}})}/2} + {\sum\limits_{j = 1}^{n}{b_{j}z_{j}}}}}} & (9)\end{matrix}$

Subsequently, the following equation (10) is induced from equation (9).

$\begin{matrix}{{{E\left\lbrack e^{C} \right\rbrack}^{\sum\limits_{j = 1}^{n}{c_{j}z_{j}}}} = {{{E\left\lbrack e^{A} \right\rbrack}^{\sum\limits_{j = 1}^{n}{a_{j}z_{j}}}} + {{E\left\lbrack e^{B} \right\rbrack}^{\sum\limits_{j = 1}^{n}{b_{j}z_{j}}}}}} & (10)\end{matrix}$

Accordingly, when the coefficient c_(j) may be determined irrespectivelyof the arbitrary supplemental lognormal distribution Z, and thusequation (10) may become an identical equation with respect to thedistribution Z, the polynomials for the third lognormal distributione^(C) may be determined irrespective of the added lognormaldistributions.

However, as equation (10) may be very complicated, modified equations ofequation (10) may be used for determining the c_(j).

For example, equation (10) may be expanded into a first order Taylorseries and the Taylor series may be arranged with respect to the Z oncondition that equation (10) may be an identical equation with respectto the distribution Z, as shown in the following equation (11).

$\begin{matrix}{{{E\left\lbrack e^{C} \right\rbrack}{^{\sum\limits_{j = 1}^{n}{c_{j}z_{j}}}\left( {\sum\limits_{j = 1}^{n}{c_{j}\left( {z_{j} - k_{j}} \right)}} \right)}} = {{{E\left\lbrack e^{A} \right\rbrack}{^{\sum\limits_{j = 1}^{n}{a_{j}z_{j}}}\left( {\sum\limits_{j = 1}^{n}{a_{j}\left( {z_{j} - k_{j}} \right)}} \right)}} + {{E\left\lbrack e^{B} \right\rbrack}{^{\sum\limits_{j = 1}^{n}{b_{j}z_{j}}}\left( {\sum\limits_{j = 1}^{n}{b_{j}\left( {z_{j} - k_{j}} \right)}} \right)}}}} & (11)\end{matrix}$

In the above equation (11), k_(j) indicates an average of thecoefficients of the lognormal distribution. The third lognormaldistribution e^(C) may include a summation of the actual leakagecurrents in the same cell, and thus is the same as the average of thecoefficients of the first and second lognormal distributions e^(A) ande^(B).

The c_(j) of the identical equation (11) may be determined as thefollowing equation (12) irrespective of the Z.

$\begin{matrix}{{c_{j} = \frac{{{E\left( e^{A} \right)}^{\sum\limits_{j = 1}^{n}{a_{j}k_{j}}}} + {{E\left( e^{B} \right)}^{\sum\limits_{j = 1}^{n}{b_{j}k_{j}}}}}{{E\left( e^{C} \right)}^{\sum\limits_{j = 1}^{n}{c_{j}k_{j}}}}}{{{{Since}\mspace{14mu} {E\left\lbrack e^{C} \right\rbrack}} = {{E\left\lbrack e^{A} \right\rbrack} + {E\left\lbrack e^{B} \right\rbrack}}},}} & (12) \\{^{\sum\limits_{j = 1}^{n}{c_{j}k_{j}}} = \frac{{{E\left( e^{A} \right)}^{\sum\limits_{j = 1}^{n}{a_{j}k_{j}}}} + {{E\left( e^{B} \right)}^{\sum\limits_{j = 1}^{n}{b_{j}k_{j}}}}}{E\left( e^{C} \right)}} & (13)\end{matrix}$

Therefore, the coefficients c₁ to c_(n) may be determined by equation(13) and the coefficient c_(n+1) may be determined by the followingequation (14).

$\begin{matrix}{c_{n + 1} = \sqrt{\sigma_{c}^{2} - {\sum\limits_{j = 1}^{n}c_{j}^{2}}}} & (14)\end{matrix}$

Accordingly, the third lognormal distribution e^(C), which may be anarithmetic summation of the first and second lognormal distributionse^(A) and e^(B), may be expressed as exponential polynomialsirrespective of the arbitrarily added lognormal distribution e^(Z).

As a result, various actual leakage currents in a cell of the wafer chip100 may be arithmetically summed up step by step to thereby generate asingle virtual equivalent leakage current of the cell, which may beexpressed as the single virtual leakage characteristic function of thecell. Therefore, the equivalent leakage current may include all kinds ofthe actual leakage currents generated from the cell and the virtualleakage characteristic function may be statistically equivalent to allkinds of the actual leakage characteristic functions. Hereinafter, thecell of which the leakage characteristics is determined by the virtualleakage characteristic function is referred to as virtual cell.

Since the virtual leakage characteristic function may be obtained byarithmetic summation of the actual leakage characteristic functions, aneffect of an arbitrary actual leakage current to the virtual equivalentleakage current of the virtual cell may be easily estimated according tothe present example embodiment of the present inventive concept.

Supposing that an arbitrary virtual cell i of which the virtual leakagecharacteristic function is determined may include three kinds of theactual leakage characteristic functions A, B and C and another arbitraryvirtual cell j may include three kinds of the actual leakagecharacteristic functions A, B and D, the virtual leakage characteristicfunction of the virtual cell j may easily be obtained by the followingarithmetic operations.

ƒ_(total))_(celli)=ƒ(A)+ƒ(B)+ƒ(C)   (15)

ƒ_(total))_(cellj)=ƒ_(total))_(celli)−ƒ(C)+ƒ(D)   (16)

Accordingly, when the virtual leakage characteristic function of a firstcell may be known, the virtual leakage characteristic function of asecond cell having the known virtual leakage characteristic function orsimilar leakage characteristic function thereto as one of the actualleakage characteristic function thereof may be easily obtained merely byarithmetic operation of other actual leakage characteristic functionsthat are not common to both of the first and second cells. In thepresent embodiment, the actual leakage characteristic functions C and Dare not common to both of the first and second cells and the actualleakage characteristic function C is subtracted from the known actualleakage characteristic function, which is the virtual leakagecharacteristic function of the first cell, and the actual leakagecharacteristic function D may be added to the known actual leakagecharacteristic function. Therefore, when the actual leakagecharacteristic functions of a cell may be similar to those of anothercell, the virtual leakage characteristic function of the cell may beeasily obtained with a remarkably reduced computational load.

In a modified example embodiment of the present example, the abovearithmetic generation of the equivalent leakage characteristic functionmay also be used for an expectation model for expecting an effect of asupplemental actual leakage characteristic function on a virtual leakagecharacteristic function of an arbitrary virtual cell.

Supposing that an arbitrary virtual cell i of which the virtualequivalent leakage characteristic function is already known includethree kinds of the actual leakage characteristic functions A, B and Cand a supplemental actual leakage characteristic function D be added tothe virtual cell i, the virtual equivalent leakage characteristicfunction of the arbitrary virtual cell i may be easily obtained merelyby arithmetically adding the supplemental actual leakage characteristicfunction D to the known virtual equivalent leakage characteristicfunction, as shown in the following equations (17) and (18).

ƒ_(total))_(original)=ƒ(A)+ƒ(B)+ƒ(C)   (17)

ƒ_(total))_(afteradding)=ƒ_(total))_(original)+ƒ(D)   (18)

In addition, when some of the actual leakage currents may be varied in acell of the wafer chip 100, the variation of the equivalent leakagecharacteristic function of the virtual cell due to the variation of theactual leakage characteristic function may be easily estimated merely bythe arithmetic treatment of the variation of the actual leakagecharacteristic function. That is, a variation analysis of the actualleakage characteristic function, which is substantially identical to anincremental analysis of the conventional Wilkinson's method, may beeasily performed with a remarkably reduced computational load.

Furthermore, the variation analysis of the actual leakage characteristicfunction may be applied to a cell incremental analysis for analyzing thevariation of the equivalent leakage characteristic function in case thatthe virtual cell may be added or deleted to/from the whole wafer chip100. When each of the virtual leakage characteristic functions isdetermined with respect to all of the cells of the wafer chip 100 (i.e.,a full chip leakage characteristic function may be determined on thewafer chip 100) and some of the cells may be deleted or changed inaccordance with the requirements of the circuit design, the variation ofthe full chip leakage characteristic function due to the cell change maybe easily obtained merely by the arithmetic operation without anyre-calculation of the full chip leakage characteristic function inconsideration of the cell change. Therefore, the variation of the fullchip leakage characteristic function due to the cell change may beeasily estimated in designing an electric circuit without any additionalcomputational load.

A number of the virtual cell leakage characteristic functions may bearithmetically added to one another in the same segment, to therebygenerate a segment leakage characteristic function (step S400).

When a single cell is positioned in the segment A, the virtual cellleakage characteristic function may function as the segment leakagecharacteristic function and when a number of cells may be positioned inthe segments A, the arithmetic summation of the virtual cell leakagecharacteristic functions may function as the segment leakagecharacteristic function.

Since the virtual cell leakage characteristic function may be describedas an exponential polynomial of the lognormal distribution of which thecoefficients are determined not by a statistical process but by anarithmetic process, the segment leakage characteristic function may alsobe obtained by the same arithmetic summation of the virtual cell leakagecharacteristic functions. Thus, detailed descriptions on the arithmeticsummation process for adding the virtual cell leakage characteristicfunctions will be omitted.

However, when the actual leakage currents may be cased by differentprocess parameters, respectively, the actual leakage currents may not berepresented as a single random variable, and thus the virtual cellleakage characteristic function may not be equivalent with the actualleakage currents. Therefore, the actual leakage currents concerningdifferent respective process parameters may not be arithmetically addedto one another as an exponential polynomial even though the actualleakage currents may be generated in the same segment A.

For example, when two different processes may be performed on a firstcell and a second cell, respectively, the threshold voltage of the firstcell may be independent from that of the second cell, and thus thedifferent threshold voltages of the first and second cells cannot betreated as a single random variable. Therefore, the leakage currentsgenerated from the first cell and second cell may not be arithmeticallyadded to each other by the same process as described above using anexponential polynomial. However, only if the first and the second cellsmay be under the same process conditions, the actual leakage currentsmay be arithmetically added to each other although the physicalmechanisms of the actual leakage currents may be different from eachother like a sub-threshold leakage current and a gate leakage current.

Then, the segment leakage currents may be statistically added to oneanother, to thereby generate a single full chip leakage characteristicfunction dominating an overall leakage current generated from the wholewafer chip 100 (step S500).

In an example embodiment, the full chip leakage characteristic functionmay be generated by a statistical process to thereby increase theaccuracy of the leakage characteristic function. For example, a numberof the segment leakage characteristic functions may be added to oneanother by the conventional Wilkinson's method, to thereby estimate thePDF of the full chip leakage characteristic function.

In case that the full chip leakage characteristic function may begenerated by the statistical process, an average and variation of thefull chip leakage characteristic function may be determined byWilkinson's method using the average and variation of the segmentleakage characteristic function. Accordingly, the full chip leakagecharacteristic function may be determined as a statistical distribution,not as an exponential polynomial. A total leakage current may besufficiently estimated by using the statistical distribution of the fullchip leakage characteristic function.

The complexity of Wilkinson's method for determining the full chipleakage characteristic function may be varied according to the number ofthe virtual cells. However, various cells in the same wafer chip 100 maybe substantially under the same process conditions. Therefore, supposingthat an ignorable number of the cells may be under different processparameters, the complexity of Wilkinson's method for generating the fullchip leakage characteristic function may be substantially approximatedto the number of the segments A of the grid model.

As a result, the complexity of Wilkinson's method for generating thefull chip leakage characteristic function may be remarkably reduced incompared with the conventional case where each of the actual leakagecharacteristic functions may be treated by Wilkinson's method.

According to some example embodiments, lognormal distributions about aleakage characteristic function may be summed up not by a statisticalprocess but by an arithmetic process using an exponential polynomial ofthe PDF of the lognormal distribution. Therefore, the complexity ofWilkinson's method for generating a full chip leakage characteristicfunction may be remarkably reduced without any deterioration of theaccuracy of the full chip leakage characteristic function.

In addition, when performing a principal component analysis (PCA) on awafer chip, a computational overload for processing a spatialcorrelation matrix may be sufficiently minimized. Furthermore, avariation analysis for analyzing the effect of the variation of anactual leakage characteristic function on a virtual cell leakagecharacteristic function, which may be known as an incremental analysisin the conventional Wilkinson's method, may be performed merely by aseries of arithmetic operations, to thereby sufficiently reduce thecomputational loads of the conventional Wilkinson's method.

The foregoing is illustrative of example embodiments and is not to beconstrued as limiting thereof. Although a few example embodiments havebeen described, those skilled in the art will readily appreciate thatmany modifications are possible in the example embodiments withoutmaterially departing from the novel teachings and advantages of thepresent invention. Accordingly, all such modifications are intended tobe included within the scope of the present invention as defined in theclaims. In the claims, means-plus-function clauses are intended to coverthe structures described herein as performing the recited function andnot only structural equivalents but also equivalent structures.Therefore, it is to be understood that the foregoing is illustrative ofvarious example embodiments and is not to be construed as limited to thespecific example embodiments disclosed, and that modifications to thedisclosed example embodiments, as well as other example embodiments, areintended to be included within the scope of the appended claims.

1. A method of estimating a leakage current in semiconductor device,comprising: dividing a chip on a substrate into a number of segments,the chip including a plurality of cells on which various unit conductivestructures of an integrated circuit is formed; determining spatialcorrelation between process parameters that are concerned with theleakage currents in each of the cells; generating a virtual cell leakagecharacteristic function of the cell by arithmetically operating actualleakage characteristic functions that determine the leakage currentsgenerated from the cell, respectively, the virtual cell leakagecharacteristic function being equivalent to actual leakagecharacteristic functions and a virtual leakage current generated by thevirtual cell leakage characteristic function being equivalent with theleakage currents in the cell; generating a segment leakagecharacteristic function by arithmetically operating the virtual cellleakage characteristic functions of each cell in the segment, thesegment leakage characteristic function determining a virtual leakagecurrent generated from all of the segments of the chip; and generating afull chip leakage characteristic function by statistically operating thesegment leakage characteristic functions of each segment in the chip,the full chip leakage characteristic function determining a virtualleakage current generated from the whole chip of the semiconductordevice.
 2. The method of claim 1, wherein the actual leakagecharacteristic function and the virtual cell leakage characteristicfunction is expressed as an exponential polynomial with respect to theprocess parameter.
 3. The method of claim 2, wherein the actual leakagecharacteristic function includes a first probability density function(PDF) determining a first leakage current caused by the processparameter and expressed as equation (1) and a second PDF determining asecond leakage current caused by the process parameter and expressed asequation (2)e^(ƒ) ¹ ^((P) ¹ _(,P) ¹ ^(. . . ,P) ^(n) )   (1)e^(ƒ) ² ^((P) ₁ ^(,P) ² ^(. . . ,P) ^(n) )   (2), and the virtual cellleakage characteristic function includes a third PDF that equalsarithmetic summation of the first and the second PDFs, and thusexpressed as equation (3),e ^(ƒ) ³ ^((P) ¹ ^(,P) ² ^(. . . ,P) ^(n) )=e ^(ƒ) ¹ ^((P) ¹ ^(,P) ²^(. . . ,P) ^(n) ⁾ +e ^(ƒ) ² ^((P) ¹ ^(,P) ² ^(. . . . ,P) ^(n) ⁾   (3)(wherein the small capital letters e in the above equations indicates anatural log and p_(i) indicates the process parameter that causes theleakage current in the cell).
 4. The method of claim 3, wherein anexponential term of the first PDF includes a polynomial of a normaldistribution having an average of a₀ and a variation of$\sum\limits_{j = 1}^{n + 1}a_{j}^{2}$ so that the first PDF isexpressed as equation (4) indicating a lognormal distribution,$\begin{matrix}{^{a_{0} + {\sum\limits_{j = 1}^{n}{a_{j}P_{j}}} + {a_{n + 1}R}},} & (4)\end{matrix}$ an exponential term of the second PDF includes apolynomial of a normal distribution having an average of b₀ and avariation of $\sum\limits_{j = 1}^{n + 1}b_{j}^{2}$ so that the secondPDF is expressed as equation (5) indicating a lognormal distribution,$\begin{matrix}{^{b_{0} + {\sum\limits_{j = 1}^{n}{b_{j}P_{j}}} + {b_{n + 1}R}},} & (5)\end{matrix}$ and an exponential term of the third PDF includes apolynomial of a normal distribution having an average of c₀ and avariation of $\sum\limits_{j = 1}^{n + 1}c_{j}^{2}$ so that the secondPDF is expressed as equation (6) indicating a lognormal distribution,$\begin{matrix}{^{c_{0} + {\sum\limits_{j = 1}^{n}{c_{j}P_{j}}} + {c_{n + 1}R}},} & (6)\end{matrix}$ on condition that${c_{0} = {{2\; {\ln \left( M_{1} \right)}} - {\frac{1}{2}{\ln \left( M_{2} \right)}}}},{{\sum\limits_{j = 1}^{n + 1}c_{j}^{2}} = {{\ln \left( M_{2} \right)} - {2\; {\ln \left( M_{1} \right)}}}},{c_{j} = \frac{{{E\left( e^{f_{1}} \right)}^{\sum\limits_{j = 1}^{n}{a_{j}k_{j}}}a_{j}} + {{E\left( e^{f_{2}} \right)}^{\sum\limits_{j = 1}^{n}{b_{j}k_{j}}}b_{j}}}{{E\left( e^{f_{3}} \right)}^{\sum\limits_{j = 1}^{n}{c_{j}k_{j}}}}},{and}$${c_{n + 1} = \sqrt{\sigma_{c}^{2} - {\sum\limits_{j = 1}^{n}c_{j}^{2}}}},$wherein a, b and c denotes a fitting coefficient of a normaldistribution, M1 and M2 denotes first and second moments of the thirdPDF, R denotes a local parameter by which the first and the secondleakage currents are generated independently from each other and Pdenotes a global parameter by which both of the first and the secondleakage currents are commonly generated.
 5. The method of claim 4,wherein the first and second moments of the third PDF is statisticallyobtained as follows:M ₁ =E[e ^(ƒ) ¹ +e ^(ƒ) ² ], M ₂ =E[(e ^(ƒ) ¹ +e ^(ƒ) ² )²].
 6. Themethod of claim 4, wherein the fitting coefficient c_(j) is obtainedthrough steps of: applying a second moment equivalence condition of alognormal distribution composition in Wilkinson's method on conditionthat an arbitrary lognormal distribution e^(Z) is added to right andleft-hand sides of equation (3), to thereby obtain an equation (7) asfollows: $\begin{matrix}{{{E\left\lbrack \left( {e^{f_{3}} + e^{Z}} \right)^{2} \right\rbrack} = {{{E\left\lbrack \left( {\left( {e^{f_{1}} + e^{f_{2}}} \right) + e^{Z}} \right)^{2} \right\rbrack}->{E\left\lbrack {e^{f_{3}}e^{Z}} \right\rbrack}} = {{{E\left\lbrack {{e^{f_{1}}e^{Z}} + {e^{f_{2}}e^{Z}}} \right\rbrack}->{{E\left\lbrack e^{f_{3}} \right\rbrack}^{\sum\limits_{j = 1}^{n}{c_{j}z_{j}}}}} = {{{E\left\lbrack e^{f_{1}} \right\rbrack}^{\sum\limits_{j = 1}^{n}{a_{j}z_{j}}}} + {{E\left\lbrack e^{f_{2}} \right\rbrack}^{\sum\limits_{j = 1}^{n}{b_{j}z_{j}}}}}}}};} & (7)\end{matrix}$ expressing equation (7) into a first order Taylor series;and expressing the Taylor series of equation (7) into an identicalequation with respect to an arbitrary random variable z of the arbitrarylognormal distribution e^(Z).
 7. The method of claim 4, wherein theglobal parameter includes a chip-based variable which may be involvedwith the leakage current by the chip and an inner-chip variable havingthe spatial correlation between the leakage currents in the chip and thelocal parameter includes the variables having no spatial correlationbetween the leakage currents in the chip.
 8. The method of claim 3,wherein the first and the second leakage currents includes one of asub-threshold leakage current and a gate leakage current, respectively.9. The method of claim 1, wherein the virtual cell leakagecharacteristic function includes a PDF of a lognormal distribution ofwhich the exponential term is a polynomial of a normal distributionhaving an average of c₀ and a variation of$\sum\limits_{j = 1}^{n + 1}c_{j}^{2}$ so that the virtual cell leakagecharacteristic function is expressed as equation (8), $\begin{matrix}^{c_{0} + {\sum\limits_{j = 1}^{m}{c_{j}P_{j}}} + {c_{m + 1}R}} & (8)\end{matrix}$ (wherein R denotes a local parameter having no spatialcorrelation between the cells in the segment and m denotes a number ofthe process parameters that are not treated as the local parameter inthe segment), and the segment leakage characteristic function isgenerated by arithmetically adding the virtual cell leakagecharacteristic function expressed by equation (8) at every cell in thesegment.
 10. The method of claim 9, wherein the segment leakagecharacteristic function includes a PDF of a lognormal distribution ofwhich the exponential term is a polynomial of a normal distribution. 11.The method of claim 10, wherein arithmetically adding the virtual cellleakage characteristic function includes: applying a second momentequivalence condition of a lognormal distribution composition inWilkinson's method, to thereby obtain an exponential polynomialequation; expressing the exponential polynomial equation into a firstorder Taylor series; and expressing the Taylor series of the exponentialpolynomial equation into an identical equation with respect to a randomvariable of an arbitrary lognormal distribution.
 12. The method of claim9, wherein the segment leakage characteristic function includes one of asub-threshold leakage current and a gate leakage current.
 13. The methodof claim 1, wherein generating the full chip leakage characteristicfunction includes obtaining an average and a variation by using firstand second moments of a number of the segment leakage characteristicfunctions.
 14. The method of claim 1, wherein the actual leakagecharacteristic function is obtained by analyzing experimental dataincluding the leakage currents and the process parameters.
 15. Themethod of claim 14, wherein analyzing the experimental data includesperforming a regression analysis process onto the experimental data, sothat a statistical relation between the leakage current and the processparameter is generated.
 16. The method of claim 14, wherein the processparameter includes a global parameter having a chip-based variable whichmay be involved with the leakage current by the chip and an inner-chipvariable which has the spatial correlation between the leakage currentsin the chip and the local parameter includes the variables having nospatial correlation between the leakage currents in the chip.
 17. Themethod of claim 16, wherein the process parameter includes a randomparameter having relation to a random variation that is randomly causedby environmental factors in performing a process and a systematicparameter having relation to a systematic variation that is caused byphysical factors of a process system for performing the process.
 18. Themethod of claim 17, wherein the random variation is expressed as a PDFhaving the random parameter as a variable which determines a probabilitydistribution and the systematic variation is expressed as a spatialcorrelation matrix.
 19. The method of claim 17, wherein the processparameter includes one of a temperature of a deposition process, athickness of a deposited layer, a pattern width and a gate voltage. 20.The method of claim 1, further comprising performing a variationanalysis by arithmetically operating the virtual cell leakagecharacteristic function and a supplemental leakage characteristicfunction that determines a supplemental leakage current generated fromthe cell, to thereby analyze variation of the virtual cell leakagecharacteristic function due to the supplemental leakage characteristicfunction.